A Spectral Splitting Theorem for the $N$-Bakry Émery Ricci tensor

We extend the spectral generalization of the Cheeger-Gromoll splitting theorem to smooth metric measure space. We show that if a complete non-compact weighted Riemannian manifold $(M,g,e^{-f}dvolg)$ of dimension $n\ge 2$ has at least two ends where $f$ is smooth and bounded. If there is some $N\in (0,\mathrm{\infty })$ and $\gamma <{(\frac{1}{(n-1)(1+\frac{n-1}{N})}+\frac{n-1}{4})}^{-1}$ such that $${\lambda }_1(-\gamma {\mathrm{\Delta }}_f+{\mathrm{Ric}}_f^N)\ge 0$$then $M$ splits isometrically as $\mathbb{R}\times X$ for some complete Riemannian manifold $X$ with $({\mathrm{Ric}}_X{)}_f^N\ge 0$. The estimate can recover the spectral splitting result and its sharp constant $\frac{4}{n-1}$ in Antonelli-Pozzetta-Xu and and Catino–Mari–Mastrolia–Roncoroni.